Effective Field Enhancement Factor Research Articles

Scaling in large area field emitters and the emission dimension

Electrostatic shielding is an important consideration for large area field emitters (LAFEs) and results in a distribution of field enhancement factors even when the constituent emitters are identical. Ideally, the mean and variance together with the nature of the distribution should characterize a LAFE. In practice, however, it is generally characterized by an effective field enhancement factor obtained from a linear fit to a Fowler–Nordheim plot of the I−V data. An alternate characterization is proposed here based on the observation that for a dense packing of emitters, shielding is large and LAFE emission occurs largely from the periphery, while well separated emitter tips show a more uniform or two-dimensional emission. This observation naturally leads to the question of the existence of an emission dimension, De, for characterizing LAFEs. We show here that the number of patches of size LP in the ON-state (above average emission) scales as N(LP)∼LP−De in a given LAFE. The exponent De is found to depend on the applied field (or voltage) and approaches De=2 asymptotically.

Statistical Analysis of Field-Emission Currents

The current due to cold tunneling of electrons from a metallic surface exposed to high electric fields, regularly named ``dark current,'' is commonly described in modern literature using an analytical approximate solution provided by Murphy and Good [Phys. Rev. 102, 1464 (1956)]. This expression, which corrects earlier work by Fowler and Nordheim, is a Fowler-Nordheim-type equation: $I\ensuremath{\sim}{E}^{2}\mathrm{exp}(\ensuremath{-}a/E)$, where $I$ is the dark current, $E$ is the local electric field, and $a$ is a system-specific constant. In this paper, a numerical approximation, rather than the analytical one given by Murphy and Good, is presented. This approximation is accurate over a wide range of fields, and is used to derive the effective field enhancement factor $\ensuremath{\beta}$. On the basis of this approximation, and considering local field and current fluctuations, two alternative methods for $\ensuremath{\beta}$ estimation are presented. These methods allow instantaneous field-specific estimation of $\ensuremath{\beta}$, rather than the average estimate derived with current methods. The applicability of fluctuation-based methods is demonstrated by numerical simulation in a variety of conditions. The methods are applied to estimate $\ensuremath{\beta}$ using fluctuation analysis in experimental measurements that were not dedicated for this purpose. An open-source code for the implementation of fluctuation-derived $\ensuremath{\beta}$ estimation is provided, with an analysis of possible future experimental opportunities using dedicated experiments.

Carbon Nanotube Fiber Field Emission Array Cathodes

Field emission cathodes made from single bulk carbon nanotube (CNT) fibers have demonstrated high emission currents and long lifetimes. This paper investigates the goal of achieving even higher current levels from field emission array (FEA) cathodes comprised of multiple CNT fibers arranged in various array configurations. Arrays were fabricated with 25- $\mu \text{m}$ -diameter CNT fibers in 2-fiber ( $1 \times 2$ ), 4-fiber ( $2 \times 2$ ), and 25-fiber ( $5 \times 5$ ) configurations, and their field emission properties were measured. The $1 \times 2$ and $2 \times 2$ FEA cathodes achieved the maximum current values that scaled to greater than 2 mA per emitter. The $5 \times 5$ FEA cathode achieved a high maximum current value of 22 mA and exhibited stable emission for 10 h at ~9 mA at an applied field strength of 0.11 V/ $\mu \text{m}$ . The Fowler–Nordheim theory was used to calculate the effective field enhancement factor ( $\beta _{\mathrm {eff}}$ ) values for all of the arrays. Electrostatic simulations were performed using COMSOL Multiphysics modeling software to model the field enhancement of a perfectly uniform $5 \times 5$ array. The $\beta _{\mathrm {eff}}$ value for the total array was compared to the individual $\beta $ values at the fiber tips predicted by a surface potential model that predicted the field amplification of all 25 CNT fibers in the array.

Modeling of stochastic processes in the emission characteristics of multitips electron sources

In this work, the modeling of the fluctuations in the current-voltage characteristics of the multi-tip field emitter was produced using adsorption/desorption theory principles. The model is based on two-component emitter with emission sites, which have two different work functions: with adsorbed particles and without ones. As a result, the cloud type fluctuations in the SK-diagram were generated and the histogram of the effective field enhancement factor was plotted. Obtained dependences was similar to the experimental ones.

Electric field distribution and current emission in a miniaturized geometrical diode

We study the electric field distribution and current emission in a miniaturized geometrical diode. Using Schwarz-Christoffel transformation, we calculate exactly the electric field inside a finite vacuum cathode-anode (A-K) gap with a single trapezoid protrusion on one of the electrode surfaces. It is found that there is a strong field enhancement on both electrodes near the protrusion, when the ratio of the A-K gap distance to the protrusion height d/h<2. The calculations are spot checked against COMSOL simulations. We calculate the effective field enhancement factor for the field emission current, by integrating the local Fowler-Nordheim current density along the electrode surfaces. We systematically examine the electric field enhancement and the current rectification of the miniaturized geometrical diode for various geometric dimensions and applied electric fields.

Effective field enhancement factor and the influence of emitted space charge

Although Fowler and Nordheim developed the basics of field emission nearly one century ago with their introduction of the Fowler-Nordheim equation (FNE), the topic continues to attract research interest particularly with the development of new materials that have been proposed as field emitters. The first order analysis of experiments typically relies upon the FNE for at minimum a basic understand of the physical emission process and its parameters of emission. The three key parameters in the FNE are the work function, emission area, and field enhancement factor, all of which can be difficult to determine under experimental conditions. This paper focuses in particular, on the field enhancement factor β. It is generally understood that β provides an indication of the surface roughness or sharpness of a field emitter cathode. However, in this paper, we experimentally and computationally demonstrate that cathodes with highly similar surface morphologies can manifest quite different field enhancements solely through having different emission regions. This fact can cause one to re-interpret results in which a single sharp emitter is proposed to dominate the emission from a field emitting cathode.

Electric field enhancement due to a saw-tooth asperity in a channel and implications on microscale gas breakdown

The electric field enhancement due to an isolated saw-tooth asperity in an infinite channel is considered with the goal of providing some inputs to the choice of field enhancement factors used to describe microscale gas breakdown. The Schwarz–Christoffel transformation is used to map the interior of the channel to the upper half of the transformed plane. The expression for the electric field in the transformed plane is then used to determine the electric field distribution in the channel as well as field enhancement near the asperity. The effective field enhancement factor is determined and its dependence on operating and geometrical parameters is studied. While the effective field enhancement factor depends only weakly on the height of the asperity in comparison to the channel, it is influenced significantly by the base angles of the asperity. Due to the strong dependence of field emission current density on electric field, the effective field enhancement factor (βeff) is shown to vary rapidly with the applied electric field irrespective of the geometrical parameters. This variation is included in the analysis of microscale gas breakdown and compared with results obtained using a constant βeff as is done traditionally. Even though results for a varying βeff may be approximately reproduced using an equivalent constant βeff independent of E-field, it might be important for a range of operating conditions. This is confirmed by extracting βeff from experimental data for breakdown in argon microgaps with plane-parallel cathodes and comparing its dependence on the E-field. While the use of two-dimensional asperities is shown to be a minor disadvantage of the proposed approach in its current form, it can potentially help in developing predictive capabilities as opposed to treating βeff as a curve-fitting parameter.

ZnO Nanowires Modified with Au Nanoparticles Exhibiting High Field-Emission Performance

Au nanoparticles are adsorbed onto the surface of vertical ZnO nanowires to fabricate a field emitter. The effective field enhancement factor (β) reaches 2.9 × 104. The nanoparticle size is below 10 nm and the diameter of ZnO nanowires is around 50 nm. Under ultraviolet light illumination, the nano-size emitter has a β value of 4.2 × 104, which is more than that of pure ZnO nanowires.

Experimental study of electric field screening by the proximity of two carbon fiber cathodes

This paper describes the first experiments that use only two carbon fiber field emitters with different separations to quantify and isolate the effect of electric field screening. Experiments show that when the separation between the two carbon fiber cathodes decreases, both the effective field enhancement factor, βeff, and the current emission decreases. For a two-emitter geometry, our experiment suggests a height of approximately 1.5 times the separation between the two cathodes as the optimum ratio to optimize the emitted current. The paper shows the analysis of the turn on voltage of the field emitters for different separations. The authors compare experimental data with Fowler–Nordheim field emission theory and particle-in-cell simulation, showing good agreement between experiment, theory, and modeling.

Demonstration of an Acid-Spun Single-Walled Nanotube Fiber Cathode

Field emission dc cold cathodes continue as an important area of research for uses such as electron microscopy, novel X-ray sources, vacuum electronic devices, terahertz sources, and high-power microwave tubes. Each of these applications typically requires high current densities with high-brightness electron beams driven by cathodes exhibiting long lifetime in the presence of deleterious conditions such as ion back bombardment and excessive heating. The Air Force Research Laboratory (AFRL) now investigates cathodes operating in dc mode for use in a terahertz traveling wave tube (TWT). The TWT requires an electron beam of 50 μm in diameter or less, at 10s of kiloelectronvolt energy with energy spreads of less than 10 eV. While AFRL has tested numerous cathodes in this regime, this paper reports on the first demonstration of a dc cathode utilizing a highly aligned carbon nanotube (CNT) rope for the electron emitter. The rope consists of individual single-walled CNTs that have been subjected to a nitrogen-enhanced acid etch and then spun into a rope configuration. Thus, the single rope emitter has an overall diameter of 100 m and a length of 1.5 mm. We report on preliminary results from this cathode, in particular the fabrication of the cathode, the dc cathode test system, and the cathode operation up to a voltage of 5 kV. The cathode operates stably to within 0.6% with a 5-mm anode-cathode gap at 5 keV and 1.0-mA current for hundreds of hours. Finally, we provide estimates of the cathode parameters such as the effective field enhancement factor (βeff) and emitting area (A) through a Fowler-Nordheim plot and comparison of the experimental data with simulations utilizing the particle-in-cell code Improved Concurrent Electromagnetic Particle-in-Cell.

Emitter spacing effects on field emission properties of laser-treated single-walled carbonnanotube buckypapers

Carbon nanotube (CNT) emitters on buckypaper were activated by laser treatment andtheir field emission properties were investigated. The pristine buckypapers and CNTemitters’ height, diameter, and spacing were characterized through optical analysis. Theemitter spacing directly impacted the emission results when the laser power and treatmenttimes were fixed. The increasing emitter density increased the enhanced fieldemission current and luminance. However, a continuous and excessive increase ofemitter density with spacing reduction generated the screening effect. As a result,the extended screening effect from the smaller spacing eventually crippled thefield emission effectiveness. Luminance intensity and uniformity of field emissionsuggest that the highly effective buckypaper will have a density of 2500 emissionspots cm − 2, which presents an effective field enhancement factor of 3721 and a moderated screeningeffect of 0.005. Proper laser treatment is an effective post-treatment process for optimizingfield emission, luminance, and durability performance for buckypaper cold cathodes.

Solution for space charge limited field emission current densities with injection velocity and geometric effects corrections

When particles are injected according to the Fowler–Nordheim (FN) field emission equation, the transmitted current density will transition to the space charge limited (SCL) current density, with increasing applied diode voltage. The actual transmitted current density is the so-called SCL-FN current density. In this work, Barbour’s analytic solution for the SCL-FN current density is modified with consideration of injection velocity and also geometric effects, by solving the advanced FN equation with the effective field enhancement factor, the energy conservation equation with an initial velocity term, and Poisson’s equation simultaneously. The solution is also extended to the relativistic regime where similar transition process is found. This solution has been verified using particle-in-cell simulation with varying diode voltage, electron injection velocity, and field enhancement factor.

One-dimensional combined field and thermionic emission model and comparison with experimental results

An integrated theoretical model has been developed to predict the entire range of emission from thermionic to field emission, including the mixed emission regime. The model assumes a Sommerfeld free electron model supply function, for which the Fermi-Dirac distribution applies with a nonzero temperature. The electron transmission coefficient is calculated in one dimension using a transfer matrix method (TMM) to solve the steady-state Schrödinger equation. Emission current densities have been measured for a periodic copper knife-edge cathode to compare with the TMM model result. It is shown that the computational result utilizing this model provides good agreement with the experimental data. Unambiguous and reliable estimates of the effective field enhancement factor βeff (βeff=Es∕Eg, where Es is the cathode surface electric field and Eg is the gap electric field between the cathode and anode) and the effective work function ϕeff are obtained from experimental measurements using this model by simultaneously fitting thermionic and field emission data for the cathode. Comparing the experimental and theoretical results reveals that finite temperature thermal contributions to the current emission can be significant in the operation of many field emission cathodes.

Field emission properties of nano-structured carbon films with carbon needles deposited on Si using high-power-density microwave-plasma CVD method

Nano-structured carbon films (NCFs) were fabricated on Si wafer chips with hydrogen–methane gas mixture by creating high-power-density microwave (MW) plasma in a quartz tube with a low-output MW power source. Carbon sheets with many nano-protuberances were complicatedly twisted each other in the center region of the NCF while carbon needles were additionally formed in the fringe region of the NCF. These NCFs yielded field emission (FE) characteristics with the FE current density of 68 mA/cm 2 at a macroscopic electric field, E, of 10 V/μm. Since the FE currents tended to be saturated even in a medium E region, no simple Fowler–Nordheim (F–N) model was applicable. A model considering the F–N mechanism, statistic effects of FE tip structures and a space-charge-limited-current effect has been applied successfully to explain all the FE data observed for E < 10 V/μm. It turned out that E-dependent parameters such as the effective total emission area and the effective field enhancement factor should be employed to obtain sufficient quantitative agreements between the model and the experiments.

On the growth of carbon nanofibers on glass with a Cr layer by inductively coupled plasma chemical vapor deposition: The effect of Ni film thickness

We have studied the effect of the thickness of catalytic Ni film for the growth of vertically aligned carbon nanofibers (VA-CNFs) on glass substrates coated with a conductive underlayer of Cr. Both the pretreatment process through which the catalytic Ni nanoparticles were formed and the growth of well-aligned CNFs were carried out in an inductively coupled plasma chemical vapor deposition (ICP-CVD) system. The VA-CNFs were characterized by scanning electron microscopy, Raman spectroscopy, as well as field emission measurements. The results of VA-CNF growth shows that as the Ni film thicknesses decrease, not only the length but also the density of the CNFs drop, although the density of catalytic Ni nanoparticles increases. The variation of CNF density with Ni film thicknesses is believed to be a result of the detachment of the CNFs from the substrate, caused by the electrostatic force produced by the plasma sheath electric field, as well as an ion-enhanced chemical etching effect due to atomic/ionic hydrogen, during the ICP-CVD growth. A field emission measurement apparatus based on a metallic probe of spherical anode structure was also constructed in this study. An electrostatic image model was employed to determine the electric field distribution on the cathode surface. Along with the standard F−N field emission model, the dependence of field emission current density on the cathode surface electric field, as well as an effective field enhancement factor, were extracted from the current-voltage measurement results. The threshold electric field (Ethreshold, for a current density of 1 mA/cm2) increases from 9.2 V/μm to 13.1 V/μm, and then drops to 11.5 V/μm for the CNFs with Ni film thicknesses of 20 nm, 30 nm, and 40 nm, respectively. The electrostatic model results also indicate that the 20 nm case has the greatest space-charge effect on the emission current, consistent with the growth results that the 20 nm case has the lowest CNF density. On the other hand, the CNF length of the 40 nm case is longer than that of the 30 nm one, while the densities are nearly the same; as a result, Ethreshold for the 30 nm case is higher.

Modeling of cold emission cathode by inclusion of combined field and thermionic emission processes

Emission currents have been measured at elevated temperatures for a periodic copper knife-edge cathode. To model the emission process of the cathode, we have combined thermionic and field emission processes. Electron tunneling is calculated using a transfer matrix method. Using this model, we accurately reproduced the experimental deviation and minimum in the ln(J∕Eg2) vs 1∕Eg plot, where Eg is the gap electric field and J is the emitted current density. This phenomenon has been widely observed but no comprehensive explanation has been put forth. Unambiguous and realistic estimates of the cathode effective field enhancement factor beta averaged (βeff=Esurface∕Egap) and effective work function Φ are obtained from experimental measurements using this model.

Prebreakdown Field Emission Current and Breakdown Mechanism of a Small Vacuum Gap

Breakdown voltage and prebreakdown current between parallel plane copper electrodes in a vacuum were measured in the gap range from 0.002 to 0.2 mm using an impulse voltage of 550×5000 μs. Breakdown field strength, effective field enhancement factor, and field emission current just prior to breakdown were ascertained as a function of gap spacing. With a decrease in the gap spacing from 0.2 mm, the field emission current just prior to breakdown increased sharply and at about 0.04 mm, it was 10 or 20 times larger than that when the gapspacing is 0.2 mm. However, in the gap range from 0.04 mm to 0.002 mm, the field emission current was maintained at about a constant value. The gap spacing dependence of the field emission current could be explained by the introduction of two ideas, which are a space charge effect by emitted electrons, and a change in the breakdown mechanism.